Lecture 6 : Kronecker Product of Schur Functions Part I

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1 CS Complexity Theory A Spring 2003 Lecture 6 : Kronecker Product of Schur Functions Part I Lecturer & Scribe: Murali Krishnan Ganapathy Abstract The irreducible representations of S n, i.e. the Specht modules are indexed by partitions λ of n. For any two partitions λ, µ of n, S λ S µ = g λµν S ν, for suitable integers g λµν. The actual values of these coefficients still eludes us. We look at a formula (admittedly messy), which gives the exact values of g λµν for simple shapes λ, µ. 6.1 Recall Before we get into the main topic, let us recall what we know about the representations of S n. Let G be any finite group, and C[G] the group algebra associated with G. This so called regular representation is special since it contains all the irreducible representations of G. Irreducible representations are in bijective correspondence with the conjugacy classes of G. Let us now focus on the case G = S n. Each conjugacy class of S n is indexed by a unique partition λ of n, where the components of λ determine the cycle structure of some member of the conjugacy class (choice of member does not affect answer). Let λ = (λ 1,..., λ k ) be a partition of n, with 0 < λ 1 λ 2 λ k. Then λ := n denotes the size of λ, l(λ) := k denotes its length and λ := l(λ) + log 2 (λ i ) the space required to represent the partition. The Ferrers diagram of λ i.e. F λ is the set of left-justified boxes with λ i squares in the i-th row (1st row = top row). By abuse of notation F λ is also denoted λ. Corresponding to each partition λ, we define a λ to be the sum of those elements of S n which leave all the rows of F λ invariant (number the boxes in F λ in the canonical order). Similarly b λ is the signed sum of those elements of S n which leave the columns of F λ invariant. Finally, let c λ = a λ b λ be the Young s symmetrizer, corresponding to λ. Then the representation S λ is the image of the endomorphism on C[S n ] which takes x xc λ. Now before we come to the characters of these irreducible representation, some definitions are in order. Definition 6.1 Put x = (x 1,..., x n ), 0 r n, λ n, µ n. (r th power symmetric function) P r (x) = i xr i, (Power symmetric functions) P λ (x) = j P λ j (x), (Anti-symmetric polynomials) A λ (X) = det x λj+n j i, (Discriminant) D(x) = A (0,...,0) (x) = i<j (x i x j ), (Schur polynomials) s λ (x) = A λ (x)/d(x), (r th Homogenous symmetric functions) h r (x) = sum of all distinct monomials of degree r, and 6-1

2 6-2 Lecture 6: Kronecker Product of Schur Functions Part I (Homogenous symmetric functions) h λ (x) = j h λ j (x) Note that since A λ is anti-symmetric, s λ is a polynomial with integer coefficients. Both s λ as well as P µ form a basis for the symmetric homogenous functions of degree n, as λ and µ range over partitions of n. Let χ be class function on S n (i.e. constant on conjugacy classes). Define the Frobenious map, F(χ) = 1 χ(µ)p µ (x) (6.1) n! µ n where χ(µ), represents the value of the character at the conjugacy class corresponding to µ. Every homogenous symmetric polynomial of degree n is of the form F(χ) for a suitable character χ. This is easily seen as follows: Start with a symmetric homogenous polynomial f of degree n. Since the {P λ } s form a basis of this space, f = c λ P λ for suitable scalars λ. Now define χ so that χ(σ) = n!/n λ c λ, where λ corresponds to the conjugacy class of σ and N λ is the number of elements in that conjugacy class. Since n!/n λ is always an integer, χ(σ) is always an integral multiple of c λ. The remarkable connection between the irreducible characters and the Schur polynomials is given by s λ (x) = F(χ λ ), where χ λ is the character corresponding to the irreducible representation S λ of S n. Hence the value of the irreducible characters of S n are the coefficients which occur when the Schur polynomials are expressed in terms of the Power symmetric functions. This relation can be inverted, and similar coefficients appear when the Power symmetric functions are expressed in terms of the Schur polynomials. Given two homogenous symmetric polynomials of degree n, f 1 and f 2, define their Kronecker product f 1 f 2 as F(χ 1 χ 2 ), where χ i = F 1 (f i ), for i = 1, 2. This definition is motivated by the following observation: With this definition, we have that s λ s µ corresponds to the character λµ of the representation S λ S µ. Moreover if s λ s µ = g λµν s ν, then g λµν gives the multiplicity of the representation S ν in S λ S µ. The reasoning is as follows: If S λ S µ = g λµν S ν, then we also have χ λ χ µ = g λµν χ ν. Then by linearity of F, we have F(χ λ χ µ ) = gλµν F(χ ν ), i.e. s λ s µ = g λµν s ν as promised. Moreover, if (, ) denotes the Hall inner product on symmetric functions, then we also have that g λµν = (s λ s µ, s ν ). 6.2 Basic Definitions In the rest of this exposition, we explore the values of g λµν when λ and µ are restricted to a small class of shapes. Definition 6.2 Let 0 t n. Then the partition (1 t, n t) is said to be a hook shape and the partition (t, n t) is said to be a two-row shape. It was known before that for partitions λ n, µ n and ν n, such that λ and µ are hook shapes, then ν g λµν 2. Similarly, if λ is a hook shape and µ is a two-row shape, then ν g λµν 3. Both results hold for an arbitrary n. This is the first result, where g λµν can be unbounded.

3 Lecture 6: Kronecker Product of Schur Functions Part I 6-3 If λ and µ are partitions of n then we write λ µ if l(λ) l(µ) and λ l(λ) p µ l(µ) p for 0 p l(λ). In other words F λ is completely inside F µ, when they are superposed with the bottom left corner box aligned. If λ µ, denote by F µ/λ the diagram of the skew shape µ/λ obtained by removing the boxes corresponding to λ from F µ. Let λ n and α = (α 1,..., α k ) be a sequence of integers such that α i = n. A decomposition of λ of type α denoted by D D k = λ, is a sequence of shapes D i = λ i /λ i 1 a skew shape and D i = α i. λ 1 λ 2 λ k = λ A column strict tableau T of shape µ/λ is a filling of F µ/λ with positive integers such that in each row, the numbers weakly increase from left to right, and in each column strictly increase from bottom to top. T is said to be standard if the entries of T are precisely 1,..., T = µ/λ. Let CS(µ/λ) denote the set of all column strict tableau s of shape µ/λ and ST (µ/λ) the set of all standard tableau s of the same shape. For T CS(µ/λ), denote by w(t ) the monomial obtained by replacing i T with x i and taking the product over all boxes, i.e. w(t ) = T i=1 xni i, where N(i) is the number of occurrences of i in T. Now, the skew Schur function s µ/λ is defined by s µ/λ (x 1, x 2,... ) = T CS(µ/λ) w(t ) (6.2) Setting λ = we get a combinatorial definition of the usual Schur functions. When λ =, we call the partition µ/λ a straight shape, to distinguish it from the more general skew shape. The n th homogenous symmetric function h n is thus defined by h n = s (n). A skew shape µ/λ is said to be a horizontal r-strip, if µ/λ = r and no two boxes of µ/λ are in the same column. For two shapes λ and µ, denote by λ µ the skew diagram obtained by joining at the corners the rightmost lowest box of F λ to the leftmost highest box of F µ. Hence we have that CS(λ µ) = CS(λ) CS(µ), and this in turn implies that s λ s µ = s λ µ. So, we can write an arbitrary product of Schur functions (of straight shapes) as the Schur function of a single skew shape. 6.3 Basic Formulae We start with some basic properties of the Kronecker product. Fact 6.3 Some results about Schur functions h n s λ = s λ, i.e. F(trivial char) = h n s (1n ) s λ = s λ, where λ is the conjugate partition of λ s λ s µ = s µ s λ = s λ s µ = s µ s λ (P + Q) R = P R + Q R g λ1λ 2λ 3 = g λπ(1) λ π(2) λ π(3), for any permutation π S 3. (Pieri s Rule) h r s λ = µ s µ, where the sum is over all µ such that µ/λ is a horizontal r-strip.

4 6-4 Lecture 6: Kronecker Product of Schur Functions Part I (Jacobi-Trudi identity) s λ = det h λj i+j 1 i,j l(λ), where h 0 = 1 and h r = 0 for r < 0. (Littlewood) s α s β s λ = γ α δ β c γδλ(s α s γ )(s β s δ ), where the c γδλ are the Littlewood- Richardson coefficients, i.e. c γδλ = (s γ s δ, s λ ). Some of the above proofs can be found in [FH, Appendix A]. Pieri s rule gives us a way to multiply Schur polynomials with a homogenous symmetric function, and the Jacobi-Trudi identity allows us to express Schur polynomials in terms of homogenous symmetric functions. One simple consequence of the Jacobi- Trudi identity is that s (k) = h k, i.e. if the partition has only one part, the Schur polynomial is the same as the corresponding homogenous symmetric function. Littlewood s result was used by Garsia and Remmel to show that (s H s K ) s D = which by induction can be used to show that D 1 +D 2 =D D 1 = H D 2 = K (s H s D1 ) (s K s D2 ) (h a1... h ak ) s D = s... s D1 Dk (6.3) D 1 + +D k =D D i =a Note that the right hand side of the above expression does not contain any and the result from which it was proved contains. This is possible because in our case, the Schur functions involved are just those with exactly one part. Hence they are just h k for suitable k, and we know that h n s λ = s λ. A naïve upper bound for the number of terms which occur in the above expansion is the number of ways in which D can be partitioned into k sets with the i th set having a i elements, which is the multi-nomial coefficient ( ) D a 1... a k. Note that the right hand size vanishes if D = ai. 6.4 A naïve algorithm At this point we know enough to give an algorithm to calculate the tensor product of two Schur functions, though it is horribly inefficient. Input: Partitions λ, µ of n. Output: {g λµν }, i.e. number of times each representation S ν occurs in S λ S µ. Notation: Let l := l(λ), m := l(µ). 1. Using the Jacobi-Trudi identity, write s λ as a polynomial in the basic homogenous symmetric functions. The number of terms in the expansion is l!. 2. Similarly write s µ as a polynomial in the basic homogenous symmetric functions. The number of terms in the expansion is m!. 3. Using (P + Q) R = (P R) + (Q R), we can write s λ s µ as the sum of l!m! terms, each of which is of the form s a1... s al s b1... s bm.

5 Lecture 6: Kronecker Product of Schur Functions Part I To compute such a term, use equation (6.3) and reduce it to computing s D1... s Dl over decompositions D D l = D := (b 1 ) (b m ), with D i = a i. Number of such terms is bounded by ( bj ) ( ) a 1... a l = n a 1... a l, since bj = µ = n. 5. Consider a typical term s D1... s Dl. Each skew shape D i is a subshape of the skew shape D. Hence each of the skew Schur function s s Di is a simple product of Schur functions with one part, i.e. homogenous symmetric functions. Each typical term becomes a product of O(n) homogenous symmetric functions. 6. At this point we have eliminated the in favor of multiplication 7. Now we can use the Pieri rule to write h a h b as a sum of Schur functions. Note that if h a h b = µ s µ, then µ can have at most two parts. If a and b are both less than n, then the number of different µ s which can contribute is O(n). 8. Repeating the above step O(n) times we can calculate each term s D1... s Dl. 9. Now collecting all the terms, we get the complete decomposition of s α s β. The step of the algorithm where we use equation (6.3) contributes ( ) n a 1... a l, which can be as bad as l n / ( ) n+1 l 1 (by Pigeon Hole Principle). This step alone makes this algorithm worse than the one we get from character theory for large l, unless we can reduce the number of terms we need to deal with by a lot. The total running time for this algorithm is O(l! m! l n ((l 1)/n) (l 1) n 2 ), assuming all polynomial operations can be done in unit time (which is not true but its cost is only 2 O (n) which is dominated by the l n any way). If one is only interested in the coefficient of one particular s ν, one may be able to improve on the running time. Conjecture: The decision problem, which given λ, µ, ν tells if g λµν > 0 is polynomial computable in the input length, i.e. in λ + µ + ν. 6.5 Outline of the proof Now we are in a position to give an outline of the proof of the main result. s (h,k) s (l,m) = ν g (h,k)(l,m)νs ν, and g (h,k)(l,m)ν = 0 if ν has more than 4 parts. Otherwise let ν = (a, b, c, d). Then g (h,k)(l,m)(a,b,c,d) is given by a sum of 8 terms each of the form U r=l 1 + min(expr 1, expr 2 ), where each L is in turn given by a max of 2 to 5 terms and each U is given by a min of 2 to 5 terms. Even though this formula is messy, it is the first formula for g λµν at least in special cases. We know proceed with the outline of the proof. The proof basically, follows the naïve algorithm of the previous section with a few tricks thrown in. From the Jacobi-Trudi identity, we have ( ) sh s s (h,k) = det k+1 = s s h 1 s h s k s h 1 s k+1 k and similarly for s (l,m). Hence we have,

6 6-6 Lecture 6: Kronecker Product of Schur Functions Part I s (h,k) s (l,m) = (s h s k s h 1 s k+1 ) (s l s m s l 1 s m+1 ) = s h s k s l s m s h s k s l 1 s m+1 s h 1 s k+1 s l s m + s h 1 s k+1 s l 1 s m+1 Let A = s h s k s l s m, B = s h s k s l 1 s m+1, C = s h 1 s k+1 s l s m and D = s h 1 s k+1 s l 1 s m+1, so that s (h,k) s (l,m) = A B C + D. Now the expansion of A, can be obtained from equation (6.3) by forming all decompositions D 1 + D 2 = (l) (m) with D 1 = h and D 2 = k, each term of which can then be evaluated by repeated applications of the Perri rule. Similar treatment can be given to the other terms as well. Let A, B, C, D represent the set of configurations which occur in the expansion of the corresponding term. The main observation, is that one can define a map I on A B C D to itself such that I 2 = Id. Moreover the map is such that for any configuration S for which I(S) S, the terms in the expansion of s (h,k) s (l,m) for S and I(S) cancel each other out. Hence it will be enough to look at those configurations for which I(S) = S. References [FH] William Fulton and Joe Harris, Representation Theory, A First Course. Springer Verlag [RW] J B Remmel and T Whitehead, On the Kronecker Product of Schur Functions of Two Row Shapes. Bull. Belg. Math. Soc. (1), pp , [LW] D E Littlewood, The Kronecker Product of Symmetric Group Representations. J. London Math. Soc. (1) 31, pp 89 93, 1956.